The Beat [[Frequency]] of a sound [[Wave|wave]] is the frequency of how much a pure sin-wave is oscillating. This property emerges when two waves with similar ampltiude but different frequencies interface with one another. ![[Pasted image 20251103133340.png|invert_Sepia|400]] Imagine two waves $D_1$ and $D_2$ with approximentaly the same [[Amplitude]] $A$, and their [[Superposition]] $D$. $\huge \begin{align} D_{1}&= A \sin (\omega_{1} t ) \\ D_{2}&=A\sin(\omega_{2}t) \\ D&=D_{1}+D_{2}\\ &= A\pa{ \sin(\omega_{1} t) + \sin(\omega_{2}t) }\\ \sin \theta_{1} + \sin \theta_{2} &= 2\sin\pa{\frac{\theta_{1}+\theta_{2}}{2}}\cos\pa{\frac{\theta_{1}-\theta_{2}}{2}}\\ D &= 2A \sin\left( \underbrace{\frac{\omega_{1}+\omega_{2}}{2}}_{\text{Avg}(\omega_{1},\omega_{2})} \right) \cos\pa{ \underbrace{\frac{\omega_{1}-\omega_{2}}{2}}_{\Delta \omega} } \end{align}$ In this context, the average of two the frequencies is what we would pick up as the frequency, and the difference $\Delta \omega$ would be the [[Beat Frequency]]. Note that the [[Sign|sign]] of this frequency does not matter as $\cos$ is an [[Even Function|even function]]. >[!example] > An example where $\omega_{1}=0.9$ and $\omega_{2}=8.8$ >![[Pasted image 20251103132755.png]] > >In this context, the beat frequency $\Delta \omega=7.9$